sieg ram08

Developmental Science 11:5 (2008), pp 655661 DOI: 10.1111/j.1467-7687.2008.00714.x

2008 The Authors. Journal compilation 2008 Blackwell Publishing Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.

Blackwell Publishing Ltd

SPECIAL SECTION: THE DEVELOPMENT OF MATHEMATICAL COGNITION

Playing linear numerical board games promotes low-income childrens numerical development

Robert S. Siegler and Geetha B. Ramani

Department of Psychology, Carnegie Mellon University, USA

Abstract

The numerical knowledge of children from low-income backgrounds trails behind that of peers from middle-incomebackgrounds even before the children enter school. This gap may reflect differing prior experience with informal numericalactivities, such as numerical board games. Experiment 1 indicated that the numerical magnitude knowledge of preschoolersfrom low-income families lagged behind that of peers from more affluent backgrounds. Experiment 2 indicated that playinga simple numerical board game for four 15-minute sessions eliminated the differences in numerical estimation proficiency.Playing games that substituted colors for numbers did not have this effect. Thus, playing numerical board games offers aninexpensive means for reducing the gap in numerical knowledge that separates less and more affluent children when theybegin school.

Introduction

The mathematical understanding of children fromlow-income families trails far behind that of peers frommiddle-income families (Geary, 1994; National Assess-ment of Educational Progress, 2004). This discrepancybegins before children enter school. Preschoolers fromimpoverished backgrounds count, add, subtract, andcompare magnitudes less well than more advantagedpeers (Arnold, Fisher, Doctoroff & Dobbs, 2002; Jordan,Kaplan, Olah & Locuniak, 2006; Jordan, Levine &Huttenlocher, 1994). The early differences are related tolater ones; a meta-analysis of six large longitudinal studiesindicated that low-income kindergartners mathematicalknowledge is a strong predictor of their math achievementat ages 8, 10, and 13/14 years, a much better predictorthan the kindergartners reading, attentional capacity, orsocio-emotional functioning are of those skills at thelater ages (Duncan, Dowsett, Claessens, Magnuson,Huston, Klebanov, Pagani, Feinstein, Engel, Brooks-Gunn, Sexton, Duckworth & Japel, 2007).

The present study tests two hypotheses regardingthe discrepancy between the mathematical knowledge ofpreschoolers from lower- and higher-income backgrounds.One is that low-income preschoolers relatively poormathematical performance reflects less frequent use oflinear representations of numerical magnitude in situationsthat call for such representations. The other is thatplaying numerical board games can stimulate greater

use of appropriate representations and thus improvelow-income childrens numerical competence.

Underlying both hypotheses are previous findings thatpeople represent the magnitudes of numbers in multipleways. Within the logarithmic ruler representation (Dehaene,1997), mean estimated magnitude is a logarithmic func-tion of actual magnitude (i.e. the distance between themagnitudes of small numbers is exaggerated and thedistance between the magnitudes of larger numbers isunderstated). In contrast, within the accumulator (Brannon,Wusthoff, Gallistel & Gibbon, 2001) and linear rulerrepresentations (Case & Okamoto, 1996), mean estimatedmagnitude is a linear function of actual magnitude. Withinthe accumulator representation, variability of estimatesincreases linearly with actual magnitude; within thelogarithmic and linear ruler representations, variabilityand magnitude are unrelated. Frequency of use of therepresentations changes with age, and individuals oftenuse different representations on similar tasks (Siegler &Opfer, 2003).

Increasing reliance on linear representations seemsto play a central role in the development of numericalknowledge. Consider data on number line estimation.On this task, children are presented a series of lines witha number at each end (e.g. 0 and 1000), a third number(e.g. 37) above the line, and no other markings. The taskis to estimate the location on the line of the third number.An advantage of this task is that it transparently reflectsthe ratio characteristics of the formal number system.

Address for correspondence: Robert S. Siegler, Department of Psychology, Carnegie Mellon University, Pittsburgh, PA 15213, USA; e-mail:[email protected]

656 Robert S. Siegler and Geetha B. Ramani

2008 The Authors. Journal compilation 2008 Blackwell Publishing Ltd.

Just as 80 is twice as great as 40, the estimated locationof 80 should be twice as far from 0 as the estimatedlocation of 40. More generally, estimated magnitude onthe number line should increase linearly with actualmagnitude, with a slope of 1.00.

Studies of number line estimation indicate thatchildrens estimates often do not increase linearly withnumerical size. On 0100 number lines, kindergartnersconsistently produce estimates consistent with thelogarithmic ruler representation (e.g. the estimated positionof 15 is around the actual location of 60 on the numberline). In contrast, second graders produce estimates con-sistent with the linear ruler model (Siegler & Booth, 2004).A parallel change occurs on 01000 number lines betweensecond and fourth grades, with second graders estimatesfitting the logarithmic ruler model and fourth gradersfitting the linear ruler model (Booth & Siegler, 2006).

This age-related change in number line estimation isnot an isolated phenomenon. Children undergo parallelchanges from logarithmic to linear representations atthe same ages on numerosity estimation (generatingapproximately

N

dots on a computer screen) andmeasurement estimation (drawing a line of approxi-mately

N

units) (Booth & Siegler, 2006). Consistentindividual differences are also present on these tasks,with children in the same grade usually producing thesame pattern of estimates (e.g. linear) across the threetasks. Linearity of number line estimates also correlateswith speed of magnitude comparison (Laski & Siegler,2007), learning of answers to unfamiliar addition problems(Booth & Siegler, 2008), and overall math achievementtest scores (Booth & Siegler, 2006).

Good theoretical grounds exist for these consistentrelations between the linearity of numerical magnitudeestimates and overall math achievement. Magnitudes arecentral within numerical representations; indeed, by secondor third grade, children are unable to inhibit activationof the magnitudes of numbers even when the activationinterferes with task performance (Berch, Foley, Hill &McDonough-Ryan, 1999; Nuerk, Kaufmann, Zoppoth &Willmes, 2004). Magnitude representations also influencearithmetic performance and accuracy of estimation ofanswers to arithmetic problems (Ashcraft, 1992; Case& Sowder, 1990; Dowker, 1997; LeFevre, Greenham &Waheed, 1993). When people speak of number sense,they are usually referring to ability to judge the plausibilityof answers to numerical problems on the basis of thenumbers magnitudes (Crites, 1992; Siegel, Goldsmith &Madson, 1982).

The numerical experiences that lead children to formlinear representations are unknown. It seems likely thatcounting experience contributes, but such experience appearsinsufficient to create linear representations of numericalmagnitudes (Schaeffer, Eggleston & Scott, 1974).

If counting experience is insufficient to yield linearmagnitude representations, what numerical experiencesmight contribute? One common activity that seemsideally designed for producing linear representations is

playing numerical board games, that is, board gameswith consecutively numbered, linearly arranged, equal-sizespaces, such as

Chutes and Ladders

. As noted by Sieglerand Booth (2004), such board games provide multiplecues to both the order of numbers and the numbersmagnitudes. When a child moves a token in such a game,the greater the number that the token reaches, thegreater: (a) the distance that the child has moved thetoken, (b) the number of discrete moves the child hasmade, (c) the number of number names the child hasspoken and heard, and (d) the amount of time the moveshave taken. Thus, such board games provide a physicalrealization of the linear ruler or mental number line,hypothesized by Case and Griffin (1990) to be thecentral conceptual structure underlying early numericalunderstanding.

For many young children, everyday informal activities,including board games, provide rich experiences withnumbers, which seem likely to help them form linearrepresentations. In general, children from higher SESbackgrounds have greater opportunities at home toengage in such activities than children from lower SESbackgrounds (Case & Griffin, 1990; Tudge & Doucet,2004). These differences in experience with board gamesand other informal number-related activities seem likelyto contribute to SES-related differences in early numericalunderstanding.

The intuition that playing board games might promoteearly numerical development is not a new one. Thecurricula Project Rightstart, Number Worlds, and BigMath for Little Kids reflect the same intuition (Greenes,Ginsburg & Balfanz, 2004; Griffin, 2004). However, thosecurricula and others designed to bolster low-incomepreschoolers numerical understanding (e.g. Arnold

et al

.,2002; Starkey, Klein & Wakeley, 2004; Young-Loveridge,2004) include a variety of activities (counting, arithmetic,number comparison, board games, oneone correspond-ence games, etc.), thus precluding specification of thecontribution of any one component.

The present study was designed to test whether playinga numerical board game in and of itself would enhancelow-income childrens knowledge of numerical magnitudes.We were particularly interested in whether such experiencewould help the children generate an initial linear repre-sentation of numerical magnitude, as indexed by theirnumber line estimates. Therefore, we presented 4-year-olds from low-income backgrounds with four 15-minutesessions over a 2-week period. Children in the numericalgame group played a linear board game with squares labeled110; children in the color game group were given identicalexperience except that the squares in their board gameincluded different colors rather than different numbers.

We therefore conducted two experiments. Experiment1 examined the initial numerical estimates of low-incomeand middle-income preschoolers. Experiment 2 examinedthe effects of playing linearly arranged, numerical boardgames on low-income childrens representations ofnumerical magnitude.

Board games and numerical development 657


Experiment 1

Method

Participants

Participants were 58 preschoolers (30 males, 28 females),ranging in age from 4.0 to 5.1 years (

M

= 4.7 years,

SD

= .34). Thirty-six participants (

M

= 4.6 years,

SD

= .43,56% female, 58% African American, 42% Caucasian) wererecruited from a Head Start program and three childcarecenters, all of which served low-income urban families.Almost all (96%) of these childrens families receivedgovernment subsidies for childcare expenses. The other22 children (

M

= 4.8 years,

SD

= .30, 45% female, 77%Caucasian, 23% Asian) were recruited from a predomi-nately upper-middle-class university-run preschool. Theexperimenters were a female graduate student of Indiandescent and a female, Caucasian research assistant.

Materials

Stimuli for the number line estimation task were 20 sheetsof paper, each with a 25-cm line arranged horizontallyacross the page, with 0 just below the left end of theline, and 10 just below the right end. A number from 1to 10 inclusive was printed approximately 2 cm abovethe center of the line, with each number printed on twoof the 20 sheets.

Procedure

Children met individually with an experimenter whotold them that they would be playing a game in whichthey needed to mark the location of a number on a line.On each trial, after asking the child to identify thenumber at the top of the page (and helping if needed),the experimenter asked, If this is where 0 goes (pointing)and this is where 10 goes (pointing), where does

N

go?The numbers from 1 to 10 inclusive were presented twicein random order, with all numbers presented once beforeany number was presented twice. No feedback wasgiven, only general praise and encouragement.

Results and discussion

The ideal function relating actual and estimated magni-tudes would be perfectly linear (

R

2lin

= 1.00) with a slopeof 1.00. That would mean that estimated magnitudesrose in 1:1 proportion with actual magnitudes. Linearityand slope are conceptually, and to some degree empirically,independent. Estimates can increase linearly with a slopemuch less than 1.00, and the best fitting linear functioncan have a slope of 1.00 but with more or less variabilityaround the function. Therefore, linearity and slope wereanalyzed separately.

We computed the median estimate for each number ofchildren from middle-income backgrounds, compared

the fit of the best fitting linear function to those medians,and did the same for children from low-income back-grounds. The best fitting linear function for preschoolersfrom middle-income backgrounds fit their medianestimates considerably better than did the best fittinglinear function for peers from low-income backgrounds,

R

2lin

= .94 versus .66. Analyses of individual childrensestimates provided converging evidence. The best fittinglinear function accounted for a mean of 60% of thevariance in the estimates of individual children frommiddle-income backgrounds, but only 15% of the varianceamong children from low-income backgrounds,

t

(56) =5.38,

p

< .001,

d

= 1.49.Analyses of the slopes of the best fitting linear function

yielded a similar picture. The slopes of the best fittinglinear function to the median estimates of middle-income and lower-income preschoolers were .98 and .24,respectively. Again, analyses of individual childrensestimates provided converging evidence. The mean slopefor individual children from higher-income families washigher (and closer to the ideal slope of 1.00) than thatfor children from lower-income families (mean slopes= .70 and .26,

t

(56) = 4.28,

p

< .001,

d

= 1.16).To examine knowledge of the order of numbers of

children from higher- and lower-income backgrounds,we compared each childs estimate of the magnitude ofeach number with the childs estimate for each of theother numbers, and calculated the percent estimates thatwere correctly ordered. Children from higher-incomebackgrounds correctly ordered more of their estimatesthan did children from lower-income backgrounds, 81%versus 61%,

t

(56) = 4.73,

p

< .001,

d

= 1.28.Thus, as predicted, the estimates of preschoolers from

low-income families revealed much poorer understandingof numerical magnitudes than did the estimates of pre-schoolers from higher-income families. Estimates ofmany children from low-income backgrounds did noteven reveal knowledge of the ordering of the numbersmagnitudes; 60% of children correctly ordered fewerthan 60% of the magnitudes of these single digit numbers.

Experiment 2

Experiment 2 tested the hypothesis that an hour of expe-rience playing a simple, linearly organized numericalboard game can enhance low-income childrens knowledgeof numerical magnitudes. Children from low-incomebackgrounds were randomly assigned to play one of twoboard games. The games differed only in whether eachspace included a number or a color, and whether childrencited the number or color when moving their token. Wepredicted that playing the number-based board gamewould improve childrens estimation and their representa-tion of numerical magnitude relative to: (1) their numericalrepresentation before playing the game, and (2) therepresentations of children who played the color-basedgame.



Method

Participants

Participants were the 36 children from low-incomebackgrounds in Experiment 1. Eighteen children wererandomly assigned to the experimental condition (

M

=4.6 years,

SD

= .30, 56% female, 56% African-American,44% Caucasian). The other 18 children were randomlyassigned to the control condition (

M

= 4.7 years,

SD

=.42, 56% female, 61% African-American, 39% Caucasian).The experimenter was a female graduate student ofIndian descent.

Materials

Both board games were 50 cm long and 30 cm high; hadThe Great Race written across the top; and included 11horizontally arranged, different colored squares of equalsizes with the leftmost square labeled Start. The numericalboard had the numbers 110 in the rightmost 10 squares;the color board had different colors in those squares.Children chose a rabbit or a bear token, and on eachtrial spun a spinner to determine whether the tokenwould move one or two spaces. The number conditionspinner had a 1 half and a 2 half; the color conditionspinner had colors that matched the colors of the squareson the board.

Procedure

Children met one-on-one with an experimenter for four15-minute sessions within a 2-week period. Before each

session, the experimenter told the child that they wouldtake turns spinning the spinner, would move the tokenthe number of spaces indicated on the spinner, and thatwhoever reached the end first would win. The experi-menter also said that the child should say the numbers(colors) on the squares through which they moved theirtokens. Thus, children in the numerical-board groupwho were on a 3 and spun a 2 would say, 4, 5 as theymoved. Children in the color-board group who spun ablue would say red, blue. If a child erred, the experi-menter correctly named the numbers or colors in thesquares and then had the child repeat the numbers orcolors while moving the token. Each game lasted 24minutes; children played approximately 30 games overthe four sessions. At the beginning of the first sessionand at the end of the fourth session, the experimenteradministered the number line task, which served as thepretest and posttest.

Results and discussion

The number line estimates of children who played thenumerical board game became dramatically more linearfrom pretest to posttest (Figure 1). On the pretest, thebest fitting linear function accounted for 52% of thevariance in the median estimates for each number; onthe posttest, the best fitting linear function accountedfor 96% of the variance. In contrast, the median estimatesof children who played the color board game did notbecome more linear from pretest to posttest. The bestfitting linear function accounted for 73% of the variancein the median estimates on the pretest, versus 36% of thevariance on the posttest. Viewed from another perspective,

Figure 1 Best fitting pretest and posttest linear functions among children who played the two board games.



the two conditions did not differ in the fit of the bestfitting function on the pretest, but on the posttest, thebest fitting linear function of children who played thenumerical board game fit considerably better.

Analysis of the linearity of individual childrens estimatesprovided converging evidence. Among children whoplayed the numerical board game, the percent varianceaccounted for by the linear function increased from amean of 15% on the pretest to a mean of 61% on theposttest,

t

(17) = 6.92,

p

< .001,

d

= 1.80 (Figure 2). Incontrast, there was no pretest-posttest change amongchildren who played the color version of the game; thelinear function accounted for a mean of 18% of thevariance on both the pretest and the posttest. The twogroups did not differ on the pretest, but on the posttest,the estimates of children who played the numericalboard game were much more linear, mean

R

2lin

= .61 and.18,

t

(34) = 4.85,

p

< .001,

d

= 1.62.To provide a more intuitive sense of the number line

estimation data, the accuracy of childrens estimates onthe number line was calculated using the formula that

percent absolute error

equals:

(1)

For example, if a child was asked to estimate the locationof 5 on a 010 number line and placed the mark at thelocation that corresponded to 9, the percent absoluteerror would be 40% [(95)/10]. Thus, lower absoluteerror indicates more accurate estimates.

The accuracy data paralleled the linearity data. Amongchildren who played the numerical board game, percentabsolute error decreased from pretest to posttest (28%

vs. 20%,

t

(17) = 2.43,

p

< .05,

d

= .71). In contrast, theabsolute error of estimates among children who playedthe color board game did not change (27% vs. 28%).

Findings on the slope of estimates provided convergingevidence for the conclusion that playing the numericalboard game enhanced knowledge of numerical magnitudes.For the group median estimates, the slope generatedby children in the numerical board game conditionincreased from pretest (.24) to posttest (.87). In contrast,the slope generated by children in the color board gamecondition did not change from pretest (.28) to posttest(.21). Analyses of individual childrens slopes again pro-vided converging evidence. The mean slope for childrenwho had played the numerical board game increasedfrom pretest to posttest (.23 vs. .71,

t

(17) = 5.20;

p

< .001,

d

= 1.66). In contrast, the mean slope for children whohad played the color game did not change (.30 vs. .24).Again, no differences were present on the pretest, butthe posttest slopes of those who played the numericalboard game were higher (.71 versus .24,

t

(34) = 4.00,

p

< .001).Finally, we examined pretestposttest changes in

knowledge of the ordering of the numerical magnitudesby comparing the estimated magnitudes of all pairs ofnumbers and computing the percentage of pairs for whichthe larger numbers magnitude was estimated to begreater. Children who had played the numerical boardgame ordered correctly the magnitudes of more numberson the posttest than on the pretest (81% vs. 62%,

t

(17)= 5.57,

p

< .001). Children who had played the colorversion of the game showed no pretestposttest change(61% vs. 62% correctly ordered pairs).

Did children from lower SES backgrounds who playedthe board game catch-up to age peers from higher SESbackgrounds who participated in Experiment 1? Theanswer was yes: The posttest magnitude estimates ofthe low-income children who had played the numericalboard game were equivalent on all measures to those ofthe middle-income children in Experiment 1. Considerthe analyses of individual childrens performance. Therewas no difference in the mean fit of the linear functionto individual childrens estimates (mean

R

2

lin

= .61 and .60).There was no difference in the mean slope of individualchildrens estimates (mean slopes = .65 and .66). Therewas no difference in percent correctly ordered pairs (81%in both cases). Thus, playing numerical board gameswith an adult for four 15-minute sessions over a 2-weekperiod overcame the low-income 4-year-olds initialdisadvantage on all three measures of numerical knowledgeand rendered their performance indistinguishable fromthat of middle-income peers.

General discussion

Findings from both experiments supported the hypothesesthat motivated the study. Results of Experiment 1 indicatedthat 4-year-olds from impoverished backgrounds have

Figure 2 Mean percent variance in individual childrens pretest and posttest estimates accounted for by the linear function.

Estimate Estimated QuantityScale ofEstimates



much poorer knowledge of numerical magnitudes thanage-peers from more affluent backgrounds. Results ofExperiment 2 indicated that providing children fromlow-income backgrounds with an hour of experienceplaying board games with consecutively numbered, linearlyarranged, equal-size squares improved their knowledgeof numerical magnitudes to the point where it wasindistinguishable from that of children from upper-middle-income backgrounds who did not play the games.Playing otherwise identical non-numerical board gamesdid not have this effect.

The results also suggest a partial answer to a moregeneral theoretical question: Why does the numericalknowledge of low-income preschoolers lag so far behindthat of middle-income peers? Children with greaterexposure to numerical board games enter kindergartenwith greater intuitive knowledge of numbers (Case &Griffin, 1990; Phillips & Crowell, 1994). Although designedto promote enjoyable parentchild and childchild inter-actions, numerical board games are also well engineeredto promote numerical understanding. The present studyadds causal evidence to the previous correlational supportfor the view that playing numerical board games enhancesthe numerical understanding of children who play them.

Very recent research (Ramani & Siegler, 2008) indicatesthat the positive effects of playing numerical board gamesare not limited to improved number line estimation.Playing the game also improves the counting, numberidentification, and numerical magnitude comparisonskills of preschoolers from low-income families. Thegains endure for at least 9 weeks. The minimal cost andknowledge demands of playing the board games suggestthat they could be widely adopted in preschool andHead Start centers, and that doing so would reduce thegap between lower- and middle-income childrens earlynumerical understanding.

Acknowledgements

This research was supported by Department of EducationGrants R305H020060 and R305H050035. We would liketo thank the Allegheny Intermediate Unit Head Start,Pitcairn, PA; Carnegie-Mellon Childrens School; andSalvation Army, Eastminster Church, and AlleghenyChildcare Centers for their participation in this research.

References

Arnold, D.H., Fisher, P.H., Doctoroff, G.L., & Dobb, J. (2002).Accelerating math development in Head Start classrooms.

Journal of Educational Psychology

,

92

, 762770.Ashcraft, M.H. (1992). Cognitive arithmetic: a review of data

and theory.

Cognition

,

44

, 75106.Berch, D.B., Foley, E.J., Hill, R.J., & McDonough-Ryan, P.M.

(1999). Extracting parity and magnitude from Arabic numerals:developmental changes in number processing and mental

representation.

Journal of Experimental Child Psychology

,

74

, 286308.Booth, J.L. (2005). The importance of an accurate understanding

of numerical magnitudes. Unpublished doctoral dissertation,Carnegie Mellon University, Pittsburgh, PA.

Booth, J.L., & Siegler, R.S. (2006). Developmental and individualdifferences in pure numerical estimation.

DevelopmentalPsychology

,

41

, 189201.Booth, J.L., & Siegler, R.S. (2008). Numerical magnitude repre-

sentations influence arithmetic learning.

Child Development

,

79

, 10161031.Brannon, E.M., Wusthoff, C.J., Gallistel, C.R., & Gibbon, J.

(2001). Numerical subtraction in the pigeon: evidence for alinear subjective number scale.

Psychological Science

,

12

,238243

Case, R., & Griffin, S. (1990). Child cognitive development:the role of central conceptual structures in the developmentof scientific and social thought. In C.A. Hauert (Ed.),

Develop-mental psychology: Cognitive, perceptuo-motor, and neuropsy-chological perspectives

(pp. 193230). Amsterdam: ElsevierScience.

Case, R., & Okamoto, Y. (1996). The role of central conceptualstructures in the development of childrens thought.

Mono-graphs of the Society for Research in Child Development

,

61

(Nos. 12).Case, R., & Sowder, J.T. (1990). The development of compu-

tational estimation: a neo-Piagetian analysis.

Cognition andInstruction

,

7

, 79104.Crites, T. (1992). Skilled and less skilled estimators strategies

for estimating discrete quantities.

The Elementary SchoolJournal

,

92

, 601619.Dehaene, S. (1997).

The number sense: How the mind createsmathematics

. New York: Oxford University Press.Dowker, A. (1997). Young childrens addition estimates.

Mathematical Cognition

,

3

, 141154.Duncan, G.J., Dowsett, C.J., Claessens, A., Magnuson, K.,

Huston, A.C., Klebanov, P., Pagani, L., Feinstein, L., Engel, M.,Brooks-Gunn, J., Sexton, H., Duckworth, K., & Japel, C.(2007). School readiness and later achievement.

Develop-mental Psychology

,

43

, 14281446.Frye, D., Braisby, N., Lowe, J., Maroudas, C., & Nicholls, J.

(1989). Young childrens understanding of counting andcardinality.

Child Development

,

60

, 11581171.Geary, D.C. (1994).

Childrens mathematics development:Research and practical applications

. Washington, DC:American Psychological Association.

Greenes, C., Ginsburg, H.P., & Balfanz, R. (2004). Big mathfor little kids.

Early Childhood Research Quarterly

,

19

, 159166.Jordan, N.C., Kaplan, D., Olah, L.N., & Locuniak, M.N.

(2006). Number sense growth in kindergarten: a longitudinalinvestigation of children at risk for mathematics difficulties.

Child Development

,

77

, 153175.Jordan, N.C., Levine, S.C., & Huttenlocher, J. (1994). Develop-

ment of calculation abilities in middle- and low-incomechildren after formal instruction in school.

Journal of AppliedDevelopmental Psychology

,

15

, 223240.Laski, E.V., & Siegler, R.S. (2007). Is 27 a big number?

Correlational and causal connections among numericalcategorization, number line estimation, and numericalmagnitude comparison.

Child Development

,

76

, 17231743.LeFevre, J., Greenham, S.L., & Waheed, N. (1993). The develop-

ment of procedural and conceptual knowledge in computa-tional estimation.

Cognition and Instruction

,

11

, 95132.



National Assessment of Educational Progress (2004).

NAEP2004 trends in academic progress: Three decades of studentperformance in reading and mathematics

(PublicationNo. NCES 2004564). Retrieved 5 April 2006 from http://nces.ed.gov/pubsearch/pubsinfo.asp?pubid=2005464.

Nuerk, H.-C., Kaufmann, L., Zoppoth, S., & Willmes, K.(2004). On the development of the mental number line: more,less, or never holistic with increasing age?

DevelopmentalPsychology

,

40

, 11991211.Phillips, D., & Crowell, N.A. (Eds.) (1994).

Cultural diversityand early education: Report of a workshop

. Washington, DC:National Academy Press. Retrieved 7 April 2006, from http://Newton.nap.edu/html/earled/index.html.

Ramani, G., & Siegler, R.S. (2008). Promoting broad andstable improvements in low-income childrens numericalknowledge through playing number board games.

ChildDevelopment

,

79

, 375394.Saxe, G.B., Guberman, S.R., & Gearhart, M. (1987). Social

processes in early number development.

Monographs of theSociety for Research in Child Development

,

52

(2, SerialNo. 216).

Schaeffer, B., Eggleston, V.H., & Scott, J.L. (1974). Numberdevelopment in young children.

Cognitive Psychology

,

6,357379.

Siegel, A.W., Goldsmith, L.T., & Madson, C.M. (1982). Skillin estimation problems of extent and numerosity. Journal forResearch in Mathematics Education, 13, 211232.

Siegler, R.S., & Booth, J.L. (2004). Development of numericalestimation in young children. Child Development, 75, 428444.

Siegler, R.S., & Booth, J.L. (2005). Development of numericalestimation: a review. In J.I.D. Campbell (Ed.), Handbook ofmathematical cognition (pp. 197212). New York: PsychologyPress.

Siegler, R.S., & Opfer, J. (2003). The development of numericalestimation: evidence for multiple representations of numericalquantity. Psychological Science, 14, 237243.

Starkey, P., Klein, A., & Wakeley, A. (2004). Enhancing youngchildrens mathematical knowledge through a pre-kindergartenmathematics intervention. Early Childhood Research Quarterly,19, 99120.

Tudge, J., & Doucet, F. (2004). Early mathematical experiences:observing young Black and White childrens everydayactivities. Early Childhood Research Quarterly, 19, 2139.

Young-Loveridge, J. (2004). Effects of early numeracy of aprogram using a number books and games. Early ChildhoodResearch Quarterly, 19, 8298.

/ColorImageDict > /JPEG2000ColorACSImageDict > /JPEG2000ColorImageDict > /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 150 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 120 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.00000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict > /GrayImageDict > /JPEG2000GrayACSImageDict > /JPEG2000GrayImageDict > /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 300 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.00000 /EncodeMonoImages true /MonoImageFilter /FlateEncode /MonoImageDict > /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile (None) /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName (http://www.color.org) /PDFXTrapped /Unknown

/Description >>> setdistillerparams> setpagedevice

sieg ram08

Documents